Communications of the ACM
Quantum computation and quantum information
Quantum computation and quantum information
Introduction To Automata Theory, Languages, And Computation
Introduction To Automata Theory, Languages, And Computation
Reaction-Diffusion Computers
A new computational methodology using infinite and infinitesimal numbers
ACRI'10 Proceedings of the 9th international conference on Cellular automata for research and industry
Single-tape and multi-tape Turing machines through the lens of the Grossone methodology
The Journal of Supercomputing
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The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the relativity of mathematical languages used to describe the Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by different mathematical languages (the instruments of investigation). Together with traditional mathematical languages using such concepts as ‘enumerable sets’ and ‘continuum’ a new computational methodology allowing one to measure the number of elements of different infinite sets is used in this paper. It is shown how mathematical languages used to describe the machines limit our possibilities to observe them. In particular, notions of observable deterministic and non-deterministic Turing machines are introduced and conditions ensuring that the latter can be simulated by the former are established. The authors thank the anonymous reviewers for their useful suggestions. This research was partially supported by the Russian Federal Program “Scientists and Educators in Russia of Innovations”, contract number 02.740.11.5018.